Important Study on the ∇ Dynamic Hardy–Hilbert-Type Inequalities on Time Scales with Applications

Ahmed A El-Deeb,Omar Bazighifan,Clemente Cesarano

doi:10.3390/sym14020428

Ahmed A El-Deeb, Omar Bazighifan + Show 1 more

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https://doi.org/10.3390/sym14020428

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Journal: Symmetry	Publication Date: Feb 21, 2022
Citations: 1	License type: CC BY 4.0

Affiliation: UniNettuno University, Al Azhar University

Abstract

The main objective of the present article is to prove some new ∇ dynamic inequalities of Hardy–Hilbert-type on time scales. We present and prove very important generalized results with the help of the Fenchel–Legendre transform, submultiplicative functions, and Hölder’s and Jensen’s inequality on time scales. We obtain some well-known time scale inequalities due to Hardy–Hilbert inequalities. For some specific time scales, we further show some relevant inequalities as special cases: integral inequalities and discrete inequalities. Symmetry plays an essential role in determining the correct methods for solutions to dynamic inequalities

Highlights

Background andIntroduction to ∇-Time Scales CalculusCitation: El-Deeb, A.A.; Bazighifan, O.; Cesarano, C
The basic notion is to establish a result for a dynamic equation or a dynamic inequality where the domain of the unknown function is a so-called time scale T, which is an arbitrary closed subset of the reals R; see [3,4]
We present the Fenchel–Legendre transform that will be needed in the proof of our results

Summary

Introduction

We define the graininess functions as follows: Definition 3. Let f : T → R be a function defined on a time scale T. We introduce the nabla derivative of a function f : T → R at a point t ∈ Tκ as follows: Definition 6. Let f and g : T → R be functions that are nabla differentiable at t ∈ Tκ .

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